K. Kraus, States, E#ects, and Operations, Springer--Verlag, 1983.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and Control (PhysCon 2003) August 20 22, St. Petersburg, Russia. Electronic Mail: maxim ece.northwestern.edu 1 Introduction and background In quantum information theory [1] all admissible devices are described mathematically by means of the so called quantum operations (or quantum channels) [2, 3]. Given a complex Hilbert space H, the # algebra of all bounded operators on H. In this paper we will work primarily with finite dimensional Hilbert spaces, so that includes all linear operators on H. Given Hilbert spaces 2 , a quantum channel T is a completely positive ....

....on H. In this paper we will work primarily with finite dimensional Hilbert spaces, so that includes all linear operators on H. Given Hilbert spaces 2 , a quantum channel T is a completely positive trace preserving linear map of 2 ) All such maps admit the Kraus decomposition [3] T (#) k A k #A # k , 1) where A k : 2 are operators satisfying k A # k A k = 1H 1 . This definition of the quantum channel is formulated in the Schrodinger picture, so that the density operators on 1 are mapped to density operators on 2 . The corresponding Heisenberg picture ....

K. Kraus, States, E#ects, and Operations, Springer--Verlag, 1983.

.... positive maps, quantum operations, quantum channels, noncommutative RadonNikodym theorem Electronic Mail: maxim ece.northwestern.edu 1 Introduction In the mathematical framework of quantum information theory [18] all admissible devices are modelled by the so called quantum operations [9, 20] that is, completely positive linear contractions on the algebra of observables of the physical system under consideration. Thus it is of paramount importance to have at one s disposal a good analysis toolkit for completely positive (CP) maps. There are many useful structure theorems for CP ....

....of observables of the physical system under consideration. Thus it is of paramount importance to have at one s disposal a good analysis toolkit for completely positive (CP) maps. There are many useful structure theorems for CP maps. The two best known ones, due to Stinespring [34] and Kraus [20], are de rigueur in virtually all quantum information theoretic treatments. These theorems are signi cant because each of them states that a given map is CP if and only if it is expressible in a certain canonical form. However, in many applications we need to consider whole families of CP maps. ....

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K. Kraus, States, Eects, and Operations (Springer-Verlag, Berlin, 1983).

....the following corollary. Corollary 2.2.13 Let H and K be Hilbert spaces, and let T : B(H ) B(K ) be a completely positive map. Then there exist a Hilbert space E and a bounded map V : K E such that for all A 2 B(H ) Furthermore, if T is unital, then V is an isometry. The following result [71], which carries a great deal of signi cance in quantum information theory, is a consequence of the Stinespring theorem. We provide the proof because it is instructive, and because we will come to rely on some of the techniques used in it. Theorem 2.2.14 (the Kraus representation) Let H and K be ....

....(2.18) that the dual channel T can be extended to all trace class operators on K , because any trace class operator can be written as a complex linear combination of four density operators. Finally, after all these tedious preparations, we are ready to state and prove the result, due to Kraus [71], that any channel can be represented in the ancilla form. 21 Theorem 2.2.15 (ancilla form) Let T : B(H ) B(K ) be a channel. Then there exist Hilbert spaces F and G , a unit vector 2 G , and a unitary transformation U : F such that, for any density operator on K , T ( tr F U( ....

K. Kraus, States, Eects, and Operations (Springer-Verlag, Berlin, 1983).

....of a POVM for a joint measurement which does not satisty the JMF and thus is not experimentally realizable. 2 Quantum Theory. As background for our discussion of joint measurement and state reduction in 3, we summarize here the postulates of quantum theory which we assume. For more details see [8] [9] A quantum system S is represented by a complex Hilbert space H S , which in this paper will be finite dimensional. A state of S is represented by a density operator # on H S . An observable of S is represented by a positive operator valued measure (POVM) S. According to the measurement ....

....The partial trace operator TrP maps operators on S P to operators on S [10, p. 305] We have Tr(XY ) Tr(Y X) #s 1# p 1 s 2# p 2 # = #s 1 p 1 ##s 2 p 2 #, X# Y = X# I) I# Y ) X# Y ) s# p# = X s## Y p#, Tr(X) Tr[Tr P (X) and TrP [ X# I)Y ] XTrP (Y ) [8] [9] We can use the criterion and two of the identities to prove that if the state of S P is # , then the state of S is TrP (# ) Pr(s) Tr[ E s# I)# ] Tr TrP [ E s# I)# ] Tr[E s TrP (# ) 3 Joint measurement and state reduction. Theorem (Joint Measurement) Suppose that S P is ....

K. Kraus, States, e#ects, and operations, (Springer-Verlag, Berlin, 1983).

....realizing functions f : A B , it is worth studying the physical systems which are used to represent the elements of A and B. For a physical system, one can associate the notion of a state. Here we will not enter into details, but merely outline the basic features. For more details, see [18], 16] or [8] for instance. In this section, we study the alphabet representations according to the classical physics, and the following section is devoted to the study of representing the alphabets with quantum physics in mind. 3.1 State Set A physical system CA which is capable to represent ....

Karl Kraus: States, Eects, and Operations, Springer-Verlag (1983).

....ensure, among other things, that we are not entirely comparing apples to oranges it will make more precise what quantum and classical states have in common, as well as where they differ. This will allow us a better understanding of just what classical and quantum information theory are. Kraus [3], following Ludwig, provides a treatment that will work well for my purposes. He defines a state in terms of preparation procedures and measurement procedures on a system. The notion of preparation need not imply that only laboratory systems have states; we may view systems elsewhere in nature as ....

.... noisy quantum channels on the state of a system may be described by completely positive linear maps N , from the space B(H i ) of bounded operators on a finite dimensional input Hilbert space H i , to the space B(H o ) of bounded operators on a finite dimensional output Hilbert space H o [9] 10] [3], 11] I will sometimes use the term quantum operation for a trace nonincreasing completely positive map. Such maps have representations in terms of linear operators A i [10] 3] A(ae) X i A i aeA y i ; 2.22) with X i A y i A i I ; 2.23) equality holds in the latter when the map ....

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K. Kraus, States, Effects, and Operations, Springer-Verlag, Berlin, 1983.

.... state of a system may be described by completely positive linear maps N , from the space B(H c ) of bounded linear operators on a input SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY 2 Hilbert space H c , to the space B(H o ) of bounded linear operators on an output Hilbert space H o [25] [26], 27] In this paper, we consider only discrete channels, which we de ne as having nite dimensional input and output Hilbert spaces (the word bounded in the speci cation of the input and output spaces is redundant in the discrete case) We will sometimes use the term quantum operation for a ....

....output Hilbert spaces (the word bounded in the speci cation of the input and output spaces is redundant in the discrete case) We will sometimes use the term quantum operation for a trace nonincreasing completely positive map. Such maps have representations in terms of linear operators A i [25] [26], A( X i A i A y i ; 1) with X i A y i A i I ; 2) equality holds in the latter when the map is tracepreserving. We call the set fA i g an operator decomposition, or simply decomposition, of the operation A, and sometimes write: A fA i g (3) to indicate that fA i g is an ....

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K. Kraus, States, Eects, and Operations, Springer-Verlag, Berlin, 1983.

....tr P ( n) N z = n 1 ) 1 (nN ) N : In this expression, n j ) j = 1 if j = 0, and (n j ) j is a Cartesian component of the vector n j if j = 1; 2, or 3. Moreover, even completely general measurements, described by a positive operator valued measure [13] or POVM, fE r g, can be interpreted classically, since tr E r = Z d n w( n) tr P ( n)E r z w(rj n) 19) The expression w(rj n) is nonnegative and hence can be viewed as a classical probability of observing the outcome r given that the spin directions are n. Any ....

K. Kraus, States, Eects, and Operations, Springer-Verlag, Berlin, 1983.

....the system, the F b are just the projectors on to the eigenspaces of the observable. Such a measurement of projectors is often called projectionvalued (not to be confused with a projective measurement as defined above) I will often call the elements F b of a POVM effects, following Kraus [6]. The general form for the post measurement quantum state (density operator) conditional on obtaining the result b for a measurement of a POVM consisting of operators F b is: ae 0 b = A b (ae) X i A bi aeA y bi ; 2) where the A bi satisfy X i A y bi A bi = F b : 3) Defining A : ....

K. Kraus, States, Effects, and Operations, Springer-Verlag, Berlin, 1983.

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